Abstract
It is well known that completeness properties of sets of coherent states associated with lattices in the phase plane can be proved by using the Bargmann representation or by using the kq representation which was introduced by J. Zak. In this paper both methods are considered, in particular, in connection with expansions of generalized functions in what are called Gabor series. The setting consists of two spaces of generalized functions (tempered distributions and elements of the class S*) which appear in a natural way in the context of the Bargmann transform. Also, a thorough mathematical investigation of the Zak transform is given. This paper contains many comments and complements on existing literature; in particular, connections with the theory of interpolation of entire functions over the Gaussian integers are given.
Keywords
MathematicsCompleteness (order theory)Coherent statesPure mathematicsS transformRepresentation (politics)Connection (principal bundle)Class (philosophy)Interpolation (computer graphics)Context (archaeology)Algebra over a fieldGeneralized functionComplex planeGaussianMathematical analysisWavelet transformComputer scienceImage (mathematics)Physics
Affiliated Institutions
Related Publications
Proof of completeness of lattice states in the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>k</mml:mi><mml:mi/><mml:mi>q</mml:mi></mml:math>representation
A general set of states on a lattice in the phase plane is considered. The discrete set of von Neumann coherent states for a harmonic oscillator is a particular case of the abov...
Interpolation revisited [medical images application]
Based on the theory of approximation, this paper presents a unified analysis of interpolation and resampling techniques. An important issue is the choice of adequate basis funct...
Matching pursuits with time-frequency dictionaries
The authors introduce an algorithm, called matching pursuit, that decomposes any signal into a linear expansion of waveforms that are selected from a redundant dictionary of fun...
Decomposition of Hardy Functions into Square Integrable Wavelets of Constant Shape
An arbitrary square integrable real-valued function (or, equivalently, the associated Hardy function) can be conveniently analyzed into a suitable family of square integrable wa...
A new approach to the interpolation of sampled data
A new class of interpolation kernels that are locally compact in signal space and "almost band-limited" in Fourier space is presented. The kernels are easy to calculate and lend...
Publication Info
- Year
- 1982
- Type
- article
- Volume
- 23
- Issue
- 5
- Pages
- 720-731
- Citations
- 144
- Access
- Closed
External Links
Social Impact
Altmetric
PlumX Metrics
Social media, news, blog, policy document mentions
Citation Metrics
144
OpenAlex
Cite This
A. J. E. M. Janssen
(1982).
Bargmann transform, Zak transform, and coherent states.
Journal of Mathematical Physics
, 23
(5)
, 720-731.
https://doi.org/10.1063/1.525426
Identifiers
- DOI
- 10.1063/1.525426